The Kelly Capital Growth Investment Criterion

The following sections are included:

• Dedication
• Contents
• Preface
• List of Contributors
• Acknowledgements
• Pictures

We live in an age of instant contact through email, twitter, TV, and other modes of communication. Back in the 1700s, communication was slower and mail would take weeks or months between receipt and response. The first paper in this volume and arguably the first on log utility is Daniel Bernoulli's article, written in 1738 in Basel, Switzerland where he was professor of physics and philosophy. Bernoulli, at age 25, studied in Basel, went to St. Petersburg and then returned to Basel. He is a member of the famous family of Swiss mathematicians, who were known as the first to apply mathematical analysis to the movement of liquid bodies. His article on log utility and the St. Petersburg paradox reprinted here was translated by Dr. Louise Sommer of the American University with assistance from Karl Menger, mathematics professor at the Illinois Institute of Technology, and William J. Baumol, economics professor at Princeton University. The article was published in Econometrica in 1954…

EVER SINCE mathematicians first began to study the measurement of risk there has been general agreement on the following proposition: Expected values are computed by multiplying each possible gain by the number of ways in which it can occur, and then dividing the sum of these products by the total number of possible cases where, in this theory, the consideration of cases which are all of the same probability is insisted upon. If this rule be accepted, what remains to be done within the framework of this theory amounts to the enumeration of all alternatives, their breakdown into equi-probable cases and, finally, their insertion into corresponding classifications…

If the input symbols to a communication channel represent the outcomes of a chance event on which bets are available at odds consistent with their probabilities (i.e., “fair” odds), a gambler can use the knowledge given him by the received symbols to cause his money to grow exponentially. The maximum exponential rate of growth of the gambler's capital is equal to the rate of transmission of information over the channel. This result is generalized to include the case of arbitrary odds.

Thus we find a situation in which the transmission rate is significant even though no coding is contemplated. Previously this quantity was given significance only by a theorem of Shannon's which asserted that, with suitable encoding, binary digits could be transmitted over the channel at this rate with an arbitrarily small probability of error.

The following sections are included:

• THE SUBGOAL
• SUBGOALS AND SUBJECTIVE UTILITY
• INDIVIDUAL RISK PREFERENCE
• THE NEED FOR AN OBJECTIVE CRITERION

The following sections are included:

• Introduction
• The nature of Λ*
• The asymptotic time minimization problem
• Asymptotic magnitude problem
• Problems with finite goals in coin tossing
• REFERENCES

The following sections are included:

• INTRODUCTION
• PART I. FAVORABLE GAMES
  • BLACKJACK
  • BACCARAT
  • ROULETTE
  • THE WHEEL OF FORTUNE
  • THE STOCK MARKET
• PART II. A MATHEMATICAL THEORY FOR COMMITTING RESOURCES IN FAVORABLE GAMES
  • INTRODUCTION: COIN TOSSING
  • MINIMIZING THE PROBABILITY OF RUIN
  • THE KELLY CRITERION
  • THE ADVANTAGES OF MAXIMIZING E LOG X_n
  • A STOCK MARKET EXAMPLE
  • WARRANT HEDGING
  • PORTFOLIO SELECTION USING E LOG X
  • THE KELLY CRITERION AND DEFICIENCIES IN THE MARKOWITZ THEORY OF PORTFOLIO SELECTION
• REFERENCES

The following sections are included:

• Introduction
• Samuelson's objections to logarithmic utility
• An outline of the theory of logarithmic utility as applied to portfolio selection
• Relation to the Markowitz theory; solution to problems therein
• The theory in action: results for a real institutional portfolio
• Concluding remarks
• FOOTNOTES
• REFERENCES

This paper develops a sequential model of the individual's economic decision problem under risk. On the basis of this model, optimal consumption, investment, and borrowing-lending strategies are obtained in closed form for a class of utility functions. For a subset of this class the optimal consumption strategy satisfies the permanent income hypothesis precisely. The optimal investment strategies have the property that the optimal mix of risky investments is independent of wealth, noncapital income, age, and impatience to consume. Necessary and sufficient conditions for long-run capital growth are also given.

The following sections are included:

• INTRODUCTION
• PRELIMINARIES
• AN EXAMPLE
• BORROWING AND SOLVENCY
• THE GENERAL CASE
• SERIALLY CORRELATED YIELDS
• OPTIMAL MYOPIC POLICIES
• CONCLUDING REMARKS

The following sections are included:

• INTRODUCTION
• A TEST STATISTIC FOR THE GROWTH-OPTIMUM MODEL
• A SIMPLE TEST OF THE GROWTH-OPTIMUM MODEL'S BASIC VALIDITY
• RELATIONS BETWEEN THE GROWTH-OPTIMUM AND THE SHARPE-LINTNER MODELS
• SUMMARY AND CONCLUSIONS
• REFERENCES

In this part of the book, we present papers that formalize, generalize, and extend the early results on the Kelly strategy. Although the early papers on the Kelly optimal growth strategy contained powerful results, the topic was not significant in mainstream financial economics. Unfortunately, despite lots of good evidence, it is still not a serious part of academic financial economics. Roll (1973), for example, shows that in his data set, capital growth and mean-variance portfolios are similar. Thorp (1971) had shown that the Kelly strategies were not necessarily mean-variance efficient and Markowitz (1976) argued that the Kelly strategy was the limiting mean-variance portfolio. These papers appear later in this book. Here we take up the work of those who extended the early results and begin to consider the good and bad properties of these strategies and the sensitivity of the results to data inputs and errors…

Consider the two-person zero-sum game in which two investors are each allowed to invest in a market with stocks (X₁, X₂, …,X_m) ∼ F, where X_i ⩾ 0. Each investor has one unit of capital. The goal is to achieve more money than one's opponent. Allowable portfolio strategies are random investment policies . The payoff to player 1 for policy is . The optimal policy is shown to be , where U is a random variable uniformly distributed on [0, 2], and maximizes E In over .

Curiously, this competitively optimal investment policy is the same policy that achieves the maximum possible growth rate of capital in repeated independent investments (Breiman (1961) and Kelly (1956)). Thus the immediate goal of outperforming another investor is perfectly compatible with maximizing the asymptotic rate of return.

It will be shown that each bit of information at most doubles the resulting wealth in the general stock market setup. This information bound on the growth of wealth is actually attained for certain probability distributions on the market investigated by Kelly. The bound will be shown to be a special case of the result that the increase in exponential growth of wealth achieved with true knowledge of the stock market distribution F over that achieved with incorrect knowledge G is bounded above by D(F‖G), the entropy of F relative to G.

We ask how an investor (with knowledge of the past) should distribute his funds over various investment opportunities to maximize the growth rate of his compounded capital. Breiman (1961) answered this question when the stock returns for successive periods are independent, identically distributed random vectors. We prove that maximizing conditionally expected log return given currently available information at each stage is asymptotically optimum, with no restrictions on the distribution of the market process.

If the market is stationary ergodic, then the maximum capital growth rate is shown to be a constant almost surely equal to the maximum expected log return given the infinite past. Indeed, log-optimum investment policies that at time n look at the n-past are sandwiched in asymptotic growth rate between policies that look at only the k-past and those that look at the infinite past, and the sandwich closes as k → ∞.

We exhibit an algorithm for portfolio selection that asymptotically outperforms the best stock in the market. Let x_i = (x_i1, x_i2, …,x_im)^t denote the performance of the stock market on day i, where x_ij is the factor by which the jth stock increases on day i. Let b_i = (b_i1 , b_i2, …, b_im)^t, b_ij ≥ 0, Σ_j b_ij = 1, denote the proportion b_ij of wealth invested in the jth stock on day i. Then is the factor by which wealth is increased in n trading days. Consider as a goal the wealth that can be achieved by the best constant rebalanced portfolio chosen after the stock outcomes are revealed. It can be shown that exceeds the best stock, the Dow Jones average, and the value line index at time n. In fact, usually exceeds these quantities by an exponential factor. Let x₁, x₂, …, be an arbitrary sequence of market vectors. It will be shown that the nonanticipating sequence of portfolios yields wealth such that , for every bounded sequence x₁, x₂, …, and, under mild conditions, achieves

For a market with m assets consider the minimum, over all possible sequences of asset prices through time n, of the ratio of the final wealth of a nonanticipating investment strategy to the wealth obtained by the best constant rebalanced portfolio computed in hindsight for that price sequence. We show that the maximum value of this ratio over all nonanticipating investment strategies is V_n = [∑ 2^{−nH(n₁/n, …, n_m/n)}(n !/(n₁ ! ⋯ n_m !))]⁻¹, where H(·) is the Shannon entropy, and we specify a strategy achieving it. The optimal ratio V_n is shown to decrease only polynomially in n, indicating that the rate of return of the optimal strategy converges uniformly to that of the best constant rebalanced portfolio determined with full hindsight. We also relate this result to the pricing of a new derivative security which might be called the hindsight allocation option.

We extend the optimal strategy results of Kelly and Breiman and extend the class of random variables to which they apply from discrete to arbitrary random variables with expectations. Let F_n be the fortune obtained at the nth time period by using any given strategy and let be the fortune obtained by using the Kelly–Breiman strategy. We show (Theorem 1(i)) that is a supermartingale with and, consequently, . This establishes one sense in which the Kelly–Breiman strategy is optimal. However, this criterion for ‘optimality’ is blunted by our result (Theorem 1(ii)) that for many strategies differing from the Kelly–Breiman strategy. This ambiguity is resolved, to some extent, by our result (Theorem 2) that is a submartingale with and ; and if and only if at each time period j, 1≦j≦n, the strategies leading to F_n and are ‘the same’.

There is considerable literature on the strengths and limitations of mean-variance analysis. The basic theory and extensions of MV analysis are discussed in Markowitz [1987] and Ziemba & Vickson [1975]. Bawa, Brown & Klein [1979] and Michaud [1989] review some of its problems…

This paper considers the problem of investment of capital in risky assets in a dynamic capital market in continuous time. The model controls risk, and in particular the risk associated with errors in the estimation of asset returns. The framework for investment risk is a geometric Brownian motion model for asset prices. with random rates of return. The information filtration process and the capital allocation decisions are considered separately. The filtration is based on a Bayesian model for asset prices, and an (empirical) Bayes estimator for current price dynamics is developed from the price history. Given the conditional price dynamics, investors allocate wealth to achieve their financial goals efficiently over time. The price updating and wealth reallocations occur when control limits on the wealth process are attained. A Bayesian fractional Kelly strategy is optimal at each rebalancing, assuming that the risky assets are jointly lognormal distributed. The strategy minimizes the expected time to the upper wealth limit while maintaining a high probability of reaching that goal before falling to a lower wealth limit. The fractional Kelly strategy is a blend of the log-optimal portfolio and cash and is equivalently represented by a negative power utility function, under the multivariate lognormal distribution assumption. By rebalancing when control limits are reached, the wealth goals approach provides greater control over downside risk and upside growth. The wealth goals approach with random rebalancing times is compared to the expected utility approach with fixed rebalancing times in an asset allocation problem involving stocks, bonds, and cash.

This chapter gives an overview of current research in evolutionary finance. We mainly focus on the survival and stability properties of investment strategies associated with the Kelly rule. Our approach to the study of the wealth dynamics of investment strategies is inspired by Darwinian ideas on selection and mutation. The goal of this research is to develop an evolutionary framework for practical investment advice.

In this paper, we study the Kelly criterion in the continuous time framework building on the work of E.O. Thorp and others. The existence of an optimal strategy is proven in a general setting and the corresponding optimal wealth process is found. A simple formula is provided for calculating the optimal portfolio for a set of price processes satisfying some simple conditions. Properties of the optimal investment strategy for assets governed by multiple Ornstein-Uhlenbeck processes are studied. The paper ends with a short discussion of the implications of these ideas for financial markets.

The Kelly growth optimum approach is an attractive formula for investing. If the formula is robust and can be adapted to include realistic constraints on investing, then the practicality of the method is clear. The various papers here discuss these issues including liabilities, fractional Kelly, benchmarks, fixed mix strategies, and volatility induced wealth growth…

We study the optimal behavior of an investor who is forced to withdraw funds continuously at a fixed rate per unit time (e.g., to pay for a liability, to consume, or to pay dividends). The investor is allowed to invest in any or all of a given number of risky stocks, whose prices follow geometric Brownian motion, as well as in a riskless asset which has a constant rate of return. The fact that the withdrawal is continuously enforced, regardless of the wealth level, ensures that there is a region where there is a positive probability of ruin. In the complementary region ruin can be avoided with certainty. Call the former region the danger-zone and the latter region the safe-region. We first consider the problem of maximizing the probability that the safe-region is reached before bankruptcy, which we call the survival problem. While we show, among other results, that an optimal policy does not exist for this problem. we are able to construct explicit ∊-optimal policies, for any ∊ > 0. In the safe-region, where ultimate survival is assured, we turn our attention to growth. Among other results, we find the optimal growth policy for the investor, i.e., the policy which reaches another (higher valued) goal as quickly as possible. Other variants of both the survival problem as well as the growth problem are also discussed. Our results for the latter are intimately related to the theory of Constant Proportions Portfolio Insurance.

This paper concerns the problem of optimal dynamic choice in discrete time for an investor. In each period the investor is faced with one or more risky investments. The maximization of the expected logarithm of the period by period wealth, referred to as the Kelly criterion, is a very desirable investment procedure. It has many attractive properties, such as maximizing the asymptotic rate of growth of the investor's fortune. On the other hand, instead of focusing on maximal growth, one can develop strategies based on maximum security. For example, one can minimize the ruin probability subject to making a positive return or compute a confidence level of increasing the investor's initial fortune to a given final wealth goal. This paper is concerned with methods to combine these two approaches. We derive computational formulas for a variety of growth and security measures. Utilizing fractional Kelly strategies, we can develop a complete tradeoff of growth versus security. The theory is applicable to favorable investment situations such as blackjack, horseracing, lotto games, index and commodity futures and options trading. The results provide insight into how one should properly invest in these situations.

This paper discusses the allocation of capital over time with several risky assets. The capital growth log utility approach is used with conditions requiring that specific goals are achieved with high probability. The stochastic optimization model uses a disjunctive form for the probabilistic constraints, which identifies an outer problem of choosing an optimal set of scenarios, and an inner (conditional) problem of finding the optimal investment decisions for a given scenarios set. The multiperiod inner problem is composed of a sequence of conditional one period problems. The theory is illustrated for the dynamic allocation of wealth in stocks, bonds and cash equivalents.

Active portfolio management is concerned with objectives related to the outperformance of the return of a target benchmark portfolio. In this paper, we consider a dynamic active portfolio management problem where the objective is related to the tradeoff between the achievement of performance goals and the risk of a shortfall. Specifically, we consider an objective that relates the probability of achieving a given performance objective to the time it takes to achieve the objective. This allows a new direct quantitative analysis of the risk/return tradeoff, with risk defined directly in terms of probability of shortfall relative to the benchmark, and return defined in terms of the expected time to reach investment goals relative to the benchmark. The resulting optimal policy is a state-dependent policy that provides new insights. As a special case, our analysis includes the case where the investor wants to minimize the expected time until a given performance goal is reached subject to a constraint on the shortfall probability.

In this paper, we extend the definition of fractional Kelly strategies to the case where the investor's objective is to outperform an investment benchmark. These benchmarked fractional Kelly strategies are efficient portfolios even when asset returns are not lognormally distributed. We deduce the benchmarked fractional Kelly strategies for various types of benchmarks and explore the interconnection between an investor's risk-aversion and the appropriateness of their investment benchmarks.

This paper introduces a general market modeling framework, the benchmark approach, which assumes the existence of the numéraire portfolio. This is the strictly positive portfolio that when used as benchmark makes all benchmarked non-negative portfolios supermartingales, that is intuitively speaking downward trending or trendless. It can be shown to equal the Kelly portfolio, which maximizes expected logarithmic utility. In several ways, the Kelly or numéraire portfolio is the “best” performing portfolio and cannot be outperformed systematically by any other non-negative portfolio. Its use in pricing as numéraire leads directly to the real world pricing formula, which employs the real world probability when calculating conditional expectations. In a large regular financial market, the Kelly portfolio is shown to be approximated by well-diversified portfolios.

This chapter surveys theoretical research on the long-term performance of fixed-mix investment strategies. These self-financing strategies rebalance the portfolio over time so as to keep constant the proportions of wealth invested in various assets. The main result is that wealth can be grown from volatility. Our findings demonstrate the benefits of active portfolio management and the potential of financial engineering.

Multiperiod lifetime investment-savings optimization dates at least to Ramsey (1928). Phelps (1962) extended the model to include uncertainty while maximizing expected utility of lifetime consumption by choosing between consumption and investment in a single risky asset using an additive utility function. He obtained explicit solutions for a constant member of the isolastic utility class. Samuelson (1969) and Merton (1969) in companion articles develop, following Ramsey (1928) and Phelps (1962), in both discrete-time and continuous time, lifetime portfolio selection models where the objective function is the discounted sum of concave functions of period by period consumption. Samuelson solves the case when there are interior maxima, and shows that for isoelastic period by period utility functions u′ (C) = C^δ−1, δ < 1, the optimal portfolio decisions are independent of current wealth at each stage and independent of all consumption-savings decisions with a stationary optimal policy to invest a fixed proportion of current wealth in each period. Ziemba and Vickson (2010) review this literature and point to some queries regarding the validity of the interior maxima as discussed in problems in Ziemba and Vickson (1975, 2006).

The following sections are included:

• Introduction
• Basic Assumptions
• Solution of the Problem
• Bernoulli and Isoelastic Cases
• Conclusion
• REFERENCES

This edited and updated excerpt from the Ziemba and Vickson (1975, 2006) volume discusses various aspects of the history of optimal capital accumulation and the capital growth criterion, and supplements the introduction to Part IV of this volume.

Because the outcomes of repeated investments or gambles involve products of variables, authorities have repeatedly been tempted to the belief that, in a long sequence, maximization of the expected value of terminal utility can be achieved or well-approximated by a strategy of maximizing at each stage the geometric mean of outcome (or its equivalent, the expected value of the logarithm of principal plus return). The law of large numbers or of the central limit theorem as applied to the logs can validate the conclusion that a maximum-geometric-mean strategy does indeed make it “virtually certain” that, in a “long” sequence, one will end with a higher terminal wealth and utility. However, this does not imply the false corollary that the geometric-mean strategy is optimal for any finite number of periods, however long, or that it becomes asymptotically a good approximation. As a trivial counter-example, it is shown that for utility proportional to x^γ/γ, whenever γ ≠ 0, the geometric strategy is suboptimal for all T and never a good approximation. For utility bounded above, as when γ < 0, the same conclusion holds. If utility is bounded above and finite at zero wealth, no uniform strategy can be optimal, even though it can be that the best uniform strategy will be that of the maximum geometric mean. However, asymptotically the same level of utility can be reached by an infinity of nearby uniform strategies. The true optimum in the bounded ease involves nonuniform strategies, usually being more risky than the geometric-mean maximizer's strategy at low wealths and less risky at high wealths. The novel criterion of maximizing the expected average compound return, which asymptotically leads to maximizing of geometric mean, is shown to be arbitrary.

He who acts in N plays to make his mean log of wealth as big as it can be made will, with odds that go to one as N soars, beat me who acts to meet my own tastes for risk…

The following sections are included:

• BACKGROUND
• THE SEQUENCE OF GAMES
• ALTERNATE SEQUENCE-OF-GAMES FORMALIZATIONS
• UNENDING GAMES
• CONCLUSIONS
• APPENDIX
• REFERENCES

In January 1961, I spoke at the annual meeting of the American Mathematical Society on “Fortune's Formula: The Game of Blackjack”. This announced the discovery of favorable card counting systems for blackjack. My 1962 book Beat the Dealer explained the detailed theory and practice. The ‘optimal’ way to bet in favorable situations was an important feature. In Beat the Dealer, I called this, naturally enough, “The Kelly gambling system”, since I learned about it from the 1956 paper by John L. Kelly (Claude Shannon, who refereed the Kelly paper, brought it to my attention in November of 1960). I have continued to use it successfully in gambling and in investing. Since 1966, I've called it “the Kelly Criterion”. The rising tide of theory about and practical use of the Kelly Criterion by several leading money managers received further impetus from William Poundstone's readable book about the Kelly Criterion, Fortune's Formula. (As this title came from that of my 1961 talk, I was asked to approve the use of the title) . At a value investor's conference held in Los Angeles in May, 2007, my son reported that ‘everyone’ said they were using the Kelly Criterion…

The following sections are included:

• INTRODUCTION
• THE MAIN THEOREM
• OTHER SEPARATING FAMILIES
• QUESTIONS FOR FURTHER INVESTIGATION
• REFERENCES

Using three simple investment situations, we simulate the behavior of the Kelly and fractional Kelly proportional betting strategies over medium term horizons using a large number of scenarios. We extend the work of Bicksler and Thorp (1973) and Ziemba and Hausch (1986) to more scenarios and decision periods. The results show:

• the great superiority of full Kelly and close to full Kelly strategies over longer horizons with very large gains a large fraction of the time;
• that the short term performance of Kelly and high fractional Kelly strategies is very risky;
• that there is a consistent tradeoff of growth versus security as a function of the bet size
• determined by the various strategies; and
• that no matter how favorable the investment opportunities are or how long the finite horizon is, a sequence of bad results can lead to poor final wealth outcomes, with a loss of most of the investor's initial capital.

We summarize what we regard as the good and bad properties of the Kelly criterion and its variants. Additional properties are discussed as observations…

The criticisms of the Kelly strategy based on utility are general. If preferences are identified with wealth characteristics such as growth and tail values, then there is a utility foundation for Kelly and fractional Kelly strategies…

The following sections are included:

• Introduction
• Origins of the model
• The model and its basic properties
• Conditions for capital growth
• Relationship to other long-run investment models
• Relationship to intertemporal consumption-investment models
• Growth vs. security
• Applications
• Summary
• References

The appropriate criterion for evaluating, and hence also for properly constructing, investment portfolios whose performance is governed by an infinite sequence of stochastic returns has long been a subject of controversy and fascination. A criterion based on the expected logarithm of one-period return is known to lead to exponential growth with the greatest exponent, almost surely; and hence this criterion is frequently proposed. A refinement has been to include the variance of the logarithm of return as well, but this has had no substantial theoretical justification…

This paper provides an alternative behavioral foundation for an investor's use of power utility in the objective function and its particular risk aversion parameter. The foundation is grounded in an investor's desire to minimize the objective probability that the growth rate of invested wealth will not exceed an investor-selected target growth rate. Large deviations theory is used to show that this is equivalent to using power utility, with an argument that depends on the investor's target, and a risk aversion parameter determined by maximization. As a result, an investor's risk aversion parameter is not independent of the investment opportunity set, contrary to the standard model assumption.

The expected log utility of wealth (i.e., the growth-optimal or Kelly) criterion has been oft-studied in the management science literature. It leads to the highest asymptotic growth rate of wealth, and has no adjustable “preference parameters” that would otherwise need to be precisely “adjusted” to a specific individual's needs. But risk-control concerns led to alternative criteria that stress security against under performance over finite horizons. Large deviations theory enables a straightforward generalization of log utility's asymptotic analysis that incorporates these security concerns. The result is a power utility criterion that (like log utility) is free of an adjustable risk aversion parameter, because the latter is endogenously determined by expected utility maximization itself! A Bayesian formulation of the Occam's Razor Principle is used to illustrate the unavoidable reduction of scientific testability (i.e., the ability to more easily falsify) inherent in criterion functions that introduce additional adjustable parameters that are not directly observable.

We know that in the long run, full Kelly strategies dominate other strategies, but they are very risky short term. Hence, practical applications to long sequences of wagers are especially appropriate. Current hedge fund trading that enters and exits in a few seconds is such an application. Thorp (1960) coined the term Fortunes's Formula in his application of Kelly to the game of blackjack using his card counting system. The count provides an estimate of the mean so that with more favorable counts the player should wager more. Typically, blackjack players can play about 60+ hands per hour. So this application works well for full Kelly. But the risk is high that even after a few hours, with say a 1% edge, a player may be behind. Hence, blackjack teams typically use fractional Kelly strategies, with fractions of 0.2 to 0.8 being common. Gottlieb (1984, 1985) describes the early use of these fractional Kelly strategies…

Many racetrack bettors have systems. Since the track is a market similar in many ways to the stock market one would expect that the basic strategies would be either fundamental or technical in nature. Fundamental strategies utilize past data available from racing forms, special sources, etc. to “handicap” races. The investor then wagers on one or more horses whose probability of winning exceeds that determined by the odds by an amount sufficient to overcome the track take. Technical systems require less information and only utilize current betting data. They attempt to find inefficiencies in the “market” and bet on such “overlays” when they have positive expected value. Previous studies and our data confirm that for win bets these inefficiencies, which exist for underbet favorites and overbet longshots, are not sufficiently great to result in positive profits. This paper describes a technical system for place and show betting for which it appears to be possible to make substantial positive profits and thus to demonstrate market inefficiency in a weak form sense. Estimated theoretical probabilities of all possible finishes are compared with the actual amounts bet to determine profitable betting situations. Since the amount bet influences the odds and theory suggests that to maximize long run growth a logarithmic utility function is appropriate the resulting model is a nonlinear program. Side calculations generally reduce the number of possible bets in anyone race to three or less hence the actual optimization is quite simple. The system was tested on data from Santa Anita and Exhibition Park using exact and approximate solutions (that make the system operational at the track given the limited time available for placing bets) and found to produce substantial positive profits. A model is developed to demonstrate that the profits are not due to chance but rather to proper identification of market inefficiencies.

In a previous paper (Management Science, December 1981) Hausch, Ziemba and Rubinstein (HZR) developed a system that demonstrated the existence of a weak market inefficiency in racetrack place and show betting pools. The system appeared to make possible substantial positive profits. To make the system operational, given the limited time available for placing bets, an approximate regression scheme was developed for the Exhibition Park Racetrack in Vancouver for initial belting wealth between $2500 and $7500 and a track take of 17.1%. This paper: (I) extends this scheme to virtually any track and initial wealth level; (2) develops a modified system for mUltiple horse entries; (3) allows for multiple bets; (4) analyzes the effects of the track take and breakage on profits; (5) presents recent results using this system; and (6) considers the e1ttent of the inefficiency. i.e., how much can be bet before the market becomes efficient?

The betting strategy proposed in Hausch, Ziemba and Rubinstein (1981) and Ziemba and Hausch (1984,1987) has had considerable some success in North American place and show pools. The place pool is England is very different. This paper applies a similar strategy with appropriate modifications for places bet at British racetracks. The system or minor modifications also applies in a number of other countries such as Singapore with similar betting rules. The system appears to provide positive expectation wagers. However, with the higher track take it is not known how often profitable wagers will exist or what the long run performance might be.

The following sections are included:

• Introduction
• Theory
• Calculations
• Data
• Results
• Tests
• Concluding Remarks
• References

The growth optimal investment strategy has been shown to be highly effective for structured decision problems such as blackjack, sports betting, and high frequency trading. For securities markets, these strategies are more difficult to apply due to a variety of practical issues: structural changes in market behavior due to varying risk premium and related factors, transaction costs, operational constraints, and path dependent risk measures for many investors, including surplus risks for a defined-benefit pension plan. In addition, the standard three step approach for institutional money management does not allow for rapid changes in asset allocation — especially needed during highly turbulent periods. We modify the growth models to address downside protection, along with applying a portfolio of investment strategies — to improve diversification of the portfolio. Empirical results show the benefits of the concepts during normal and crash (2008) periods.

This paper presents an intertemporal portfolio selection model for pension funds or life insurance funds that maximizes the intertemporal expected utility of the surplus of assets net of liabilities. Following Merton (Econometrica 41 (1973) 867), it is assumed that both the asset and the liability return follow Itô processes as functions of a state variable. The optimum occurs for investors holding four funds: the market portfolio, the hedge portfolio for the state variable, the hedge portfolio for the liabilities, and the riskless asset. In contrast to Merton's result in the assets only case, the liability hedge is independent of preferences and only depends on the funding ratio. With HARA utility the investments in the state variable hedge portfolio are also preference independent. Finally, with log utility the market portfolio investment depends only on the current funding ratio.

The Sharpe ratio is a very useful measure of investment performance. Because it is based on mean-variance theory, and thus is basically valid only for quadratic preferences or normal distributions, skewed investment returns can lead to misleading conclusions. This is especially true for superior investors with many high returns. Superior investors may use capital growth wagering ideas to implement their strategies, which produces higher growth rates but also higher variability of wealth…

The Medallion Fund uses mathematical ideas such as the Kelly criterion to run a superior hedge fund. The staff of technical researchers and traders, working under mathematician James Simons, is constantly devising edges that they use to generate successful trades of various durations including many short term trades that enter and exit in seconds. The fund, whose size is in the $5 billion area, has very large fees (5% management and 44% incentive). Despite these fees and the large size of the fund, the net returns have been consistently outstanding, with a few small monthly losses and high positive monthly returns; see the histogram in Figure At. Table Al shows the monthly net returns from January 1993 to April 2005. There were only l7 monthly losses in 148 months and 3 losses in 46 quarters and no yearly losses in these 12+ years of trading in our data sample. The mean monthly, quarterly and yearly net returns, Sharpe and Symmetric Downside Sharpe ratios are shown in Table A2…

The central problem for gamblers is to find positive expectation bets. But the gambler also needs to know how to manage his money, i.e., how much to bet. In the stock market (more inclusively, the securities markets) the problem is similar but more complex. The gambler, who is now an “investor”, looks for “excess risk adjusted return”. In both these settings, we explore the use of the Kelly criterion, which is to maximize the expected value of the logarithm of wealth (“maximize expected logarithmic utility”). The criterion is known to economists and financial theorists by names such as the “geometric mean maximizing portfolio strategy”, maximizing logarithmic utility, the growth-optimal strategy, the capital growth criterion, etc. The author initiated the practical application of the Kelly criterion by using it for card counting in blackjack. We will present some useful formulas and methods to answer various natural questions about it that arise in blackjack and other gambling games. Then we illustrate its recent use in a successful casino sports betting system. Finally, we discuss its application to the securities markets where it has helped the author to make a thirty year total of 80 billion dollars worth of “bets”.

The following sections are included:

• Bibliography
• Author Index
• Subject Index